Partial Identification of the Valuation Distribution in Sequential English Auctions

Partial Identification of the Valuation Distribution in Sequential English Auctions

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Introduction

Sequential auctions are common in markets for vintage wines, paintings, used cars, and on online platforms such as eBay. Bidders in these markets decide not only how much they are willing to pay for the current object, but also whether to preserve the option to compete for later objects. This trade-off makes observed bids difficult to interpret. A bidder who stops early may have a low valuation, or may value the current item highly but prefer to wait for a future opportunity. The theoretical literature has formalized this dynamic trade-off in many ways, with different models generating different bidding predictions. Risk aversion, declining marginal values, supply uncertainty, loss aversion, and other mechanisms can each rationalize different price paths and bidding strategies. This multiplicity of models, combined with the well-documented difficulty of computing equilibria in sequential auctions, raises a fundamental question for empirical work: can we learn about the valuation distribution without committing to a specific equilibrium?

In this paper, we develop a partial identification framework for sequential English auctions, a prevalent format for sequential sales, that bounds the distribution of bidders' valuations using only two behavioral assumptions: (i) bidders do not bid above their valuations, and (ii) bidders stop bidding when the opportunity cost of winning the current auction exceeds the current-period profit. These assumptions define an incomplete model in the sense of haile2003inference (hereafter HT): they restrict bidder behavior without specifying which equilibrium is played. These assumptions are substantially weaker than those required for a full dynamic equilibrium and are compatible with a broad class of sequential-auction models. The resulting bounds are therefore robust to many forms of model uncertainty that complicate structural estimation.

Knowledge of the valuation distribution (denoted by $F$ hereafter) is central to auction design. It helps determine reserve prices and expected revenue under alternative formats tang2011bounds, kim2025searchauction, as well as the design and ordering of heterogeneous items in sequential sales elmaghraby2003importance, muramoto2016sequential, shi2022implementing. In sequential auction markets, where equilibrium-based estimates of $F$ may be misspecified, informative bounds on $F$ are practically valuable: they characterize the competitiveness of the market, inform the design of auction mechanisms, and serve as inputs to counterfactual policy analysis that is robust to equilibrium assumptions.

The paper's contributions fall into three groups. First, on identification, we extend the HT approach to sequential English auctions of heterogeneous objects. HT's static condition no longer holds in a sequential setting, since a bidder may decline to outbid a rival because the future option is more valuable than winning now. We formalize this dynamic opportunity cost and derive bounds on valuations without specifying dynamic equilibrium bidding strategies, while accommodating heterogeneous objects through the common value component. The resulting bounds on the valuation distribution are derived using an order-statistic inversion approach. When individual bidders can be tracked across auctions, the panel structure provides additional identifying information. A bidder's maximum reduced bid across all periods in which she participates is a tighter lower bound on her private value than any single-period bid, yielding a direct bound on $F$ without order-statistic inversion. We also characterize sharp bounds on $F$ using the generalized instrumental variable (GIV) framework chesher2017generalized.\footnote{It is known that HT bounds in the static case are not sharp; this extends to our setting. The GIV framework provides a sharp characterization based on random set theory without requiring constructive proof. We show that the non-sharp bounds from our main analysis are nested as special cases and identify the additional moment inequalities that tighten the identified set.}

Second, on estimation and inference, we develop a moment-condition inversion estimator that pools auctions with heterogeneous numbers of bidders into a single estimation step. In the original HT application, bounds are estimated by conditioning on $N$ via kernel smoothing, which can downweight much of the data and can produce crossing bounds when the bandwidth is too wide. More broadly, order-statistic inversion is poorly behaved in finite samples near the support boundaries, as emphasized by menzel2013large. Our approach instead solves a moment equation, where each auction contributes with a weight determined by its own number of bidders: low-competition auctions carry information in the left tail, while high-competition auctions contribute more in the right tail. This mitigates the tail degeneracy of fixed-$N$ order-statistic inversion by exploiting variation in bidder counts across auctions. We establish consistency, derive the asymptotic distribution, and use analytical standard errors to construct confidence intervals, with bootstrap intervals serving as a finite-sample benchmark. To our knowledge, this pooling approach has not been used previously in the auction literature.

Third, we demonstrate the practical value of the approach through simulation experiments, two empirical applications, and three counterfactual exercises. The simulations show that applying the static HT bounds to sequential auction data produces misspecified (crossing) bounds, while our method yields informative bounds that are valid under the maintained assumptions and performs reliably in finite samples. The two applications illustrate complementary strengths of the sequential approach: in Korean wholesale used-car auctions, where terminal auctions and long within-day sequences are observed, the sequential and terminal-period inequalities deliver tight bounds on $F$ across vehicle categories. In Cars and Bids online auctions, where listings recur continuously with no terminal period and the static HT lower bound is inapplicable, the sequential approach is the only available lower bound. Our counterfactual analyses translate the bounds on $F$ into bounds on policy quantities: the option to wait lowers first-period revenue by 8--11% in the Korean market, increasing effective competition from 8 to 20 serious bidders in Cars and Bids raises seller revenue by 40--65%, and the maximin reserve varies substantially across vehicle clusters.

Our paper connects three literatures. In sequential-auction theory and closely related auction models, the canonical predictions are a martingale price process under IPV weber1983multiobject and rising prices under affiliation milgrom1982theory, yet the “declining price anomaly” is pervasive.\footnote{This anomaly is well documented in wine ashenfelter1989auctions, ashenfelter2003auctions, art beggs1997declining, and flowers vandenberg2001winner, though the direction is not universal. raviv2006new documents rising prices in New Jersey used car auctions.} Competing explanations include risk aversion mcafee1993declining, declining marginal values bernhardt1994note, kittsteiner2004declining, buyer's options black1992winner, supply uncertainty jeitschko1999equilibrium, engelbrecht1994sequential, synergies branco1997sequential, kong2021sequential, loss aversion rosato2022loss, ambiguity ghosh2021sequential, stochastic entry hendricks2012last, said2011sequential, and item ordering effects elmaghraby2003importance, muramoto2016sequential, shi2022implementing. Equilibrium computation in this setting is technically difficult caillaud2004equilibrium, benoit2001multiple, landi2018sequential, and the bid function can change qualitatively under small perturbations. Our behavioral restrictions are compatible with many of these mechanisms because they do not require a particular equilibrium bid function.

For nonparametric estimation of auctions, the dominant approach is structural: specify an equilibrium and invert the bid function guerre2000optimal, athey2002identification, athey2007nonparametric, kim2025searchauction. Extensions address affiliated values li2002structural, unknown numbers of bidders il2014nonparametric, an2010estimating, unobserved heterogeneity krasnokutskaya2011identification, and selective entry gentry2014partial. For dynamic auctions, structural methods require solving the full dynamic programming problem jofre2003estimation, donald2006empirical, kong2021sequential, groeger2014study. For sequential English auctions specifically, brendstrup2006identification achieve point identification by exploiting changing bidder composition, and brendstrup2007non proposes a nonparametric estimator, but lamy2010identification shows the latter's identification argument fails. This identification failure further motivates partial identification, which does not require invertibility of an equilibrium bid function. In partial identification of auctions, haile2003inference propose the incomplete model for static English auctions. tang2011bounds demonstrates policy relevance by computing bounds on counterfactual revenue. Our paper extends HT to the sequential setting; to the best of our knowledge, it is the first to provide partial identification bounds for valuations in sequential English auctions.

The remainder of the paper is organized as follows. Section (ref) presents the model. Section (ref) derives bounds on the valuation distribution, characterizes sharp bounds, and discusses robustness. Section (ref) discusses estimation and inference. Section (ref) presents simulations. Section (ref) applies the method to two datasets, Korean wholesale used car auctions and online vehicle auctions from Cars and Bids. Section (ref) presents three counterfactual exercises that translate the bounds on $F$ into bounds on policy quantities (future-auction uncertainty, effective competition, and reserve-price design). Section (ref) concludes. All proofs are in the appendix.

The Incomplete Sequential Auction Model

An auctioneer holds a finite series of ascending-price auctions with a single heterogeneous unit sold in each period $k \in \{1, \ldots, K\}$. Bidders, indexed by $i \in \{1, \ldots, N\}$, enter the market before the start of the sequential auctions and remain until they win an item or the final auction concludes. Each bidder is risk-neutral with single-unit demand. Supply uncertainty is captured by $\tau_{i,k} \in [0,1]$, the probability that an item is available for sale in period $k$ from bidder $i$'s perspective.

The items are heterogeneous and imperfect substitutes. Bidder $i$'s valuation for the object in period $k$ is $v_{i,k} \equiv \phi(Z_k)\, \theta_i$, where $\phi(Z_k)$ is a common value component depending on observable item characteristics $Z_k \in \mathcal{R}_Z$ with $\phi: \mathcal{R}_Z \to \mathbb{R}_{+}$, and $\theta_i$ is bidder $i$'s private value component, drawn i.i.d.\ from a distribution $F^0(\cdot)$ supported on $[\underline{\theta}, \bar{\theta}]$ and independent of $Z_k$. Bidders know their own $\theta_i$ and $\phi(Z_k)$ but not the private values of their competitors. The total number of bidders $N$ and the number of auctions $K$ are common knowledge. Each auction follows an open ascending-price format with a fixed bid increment $\Delta \geq 0$. Bidding starts at an opening price $p_k$ set at or near the reserve price $r_k$. The auctioneer accepts monotonically increasing bids, and the auction ends when only one active bidder remains; the winner pays the final bid. Let $b_{i,k}$ denote bidder $i$'s highest bid in period $k$. We impose the same multiplicative structure on bids:

This structure provides the basis for recovering the common value component $\phi(Z_k)$ from observed bids and item characteristics.

Our incomplete model consists of two behavioral assumptions.

Assumption (ref) is standard and identical to the first assumption in HT: no rational bidder accepts a sure loss. Assumption (ref) is the dynamic analogue of HT's second assumption: losing bidders weakly prefer their continuation value to the profit from winning at the current price. We impose no further restrictions on the bidding strategy; our model permits any behavior consistent with these two assumptions.

The expected payoff from future periods after period $k$ for bidder $i$ is

where $\pi_{i,t}$ is bidder $i$'s profit from winning in period $t$.\footnote{The product $\prod_{r=k+1}^{t-1}$ equals $1$ when $t=k+1$ (an empty product).} This expected future payoff represents the opportunity cost of winning in the current period.\footnote{We define the expected payoff as the anticipated future payoff after bidder $i$ has lost in period $k$. Under Assumption (ref), the bidder weighs the profit from winning the current auction against this opportunity cost.} Importantly, we do not impose any structure on the bidder's beliefs about future prices or winning probabilities.

We are primarily interested in the primitive distribution $F^0(\cdot)$. However, if the opening price $p_k > \phi(Z_k)\underline{\theta}$ is binding in period $k$, the auction is uninformative about $F^0$ below $p_k/\phi(Z_k)$. Let $\tilde{p}_k \equiv p_k/\phi(Z_k)$ denote the opening price adjusted by the common value component. Conditional on $\tilde{p}_k$, the truncated distribution is

Values of participating bidders are i.i.d.\ draws from this truncated distribution, and the unconditional truncated distribution is

Following HT, we treat $F(\cdot)$ as the target distribution of interest, because observed bids identify only the truncated distribution. As discussed in HT, $F(\cdot)$ is sufficient for answering normative and positive questions such as solving for optimal reserve prices.

Identification of the Valuation Distribution

We first derive bounds on bidder valuations using Assumptions (ref)--(ref), then construct bounds on the valuation distribution. Throughout, $n_k$ denotes the number of participating bidders in period $k$ and $I_k$ their index set. We write $\theta^{j:n_k} \sim F_{j:n_k}(\cdot)$ for the $j$th highest order statistic of private values in period $k$, $b^{j:n_k}$ for the corresponding order statistic of bids, and $\eta^{j:n_k} \equiv b^{j:n_k}/\phi(Z_k) \sim G_{j:n_k}(\cdot)$ for the reduced-bid order statistic implied by (ref).

Bounds on Bidder Valuations

We begin with the illustrative two-period case before stating the general results. Consider $K = 2$ and set $p_1=p_2=0$ for simplicity. Under Assumption (ref), bidder $i$'s private component is bounded below by her reduced bid in every period. In the last period, there is no future auction, so Assumption (ref) implies the same result as in the static case: all participating bidders who bid below the highest bid have valuations below $(b^{1:n_2} + \Delta)/\phi(Z_2)$. In the first period, Assumption (ref) provides an additional upper bound. All bidders who bid below the highest bid have a weakly higher expected payoff from period 2 than the profit from winning in period 1:

The left-hand side is the profit from winning the first auction by raising the bid slightly above the highest bid. The second line uses $P(\cdot) \leq 1$ and $\tau_{i,2} \leq 1$, and the last line bounds the expected profit by the maximum possible gross value of the future object. Suppose that $\phi(Z_1) > \phi(Z_2)$. Rearranging yields

Now turning to the general $K$-period case, Assumption (ref) implies that valuations are bounded below by observed bids:

In the last period, Assumption (ref) reduces to HT's second assumption, since there is no future auction. Hence we derive the following upper bound:

For a non-terminal period $k < K$, Assumption (ref) implies

The key step is bounding the expected future payoff $W_k(\theta_i)$. For a retained auction $k$, we specify a continuation-value index $C_k$ and a nonnegative payment term $m_k$ such that

for the non-winning bidders. The index $C_k$ is an upper bound on the common-value component of the relevant future opportunity set, and $m_k$ is a lower bound on the payment required to realize that opportunity. Together, they give an observable upper bound on the net continuation payoff. Combining (ref) with (ref) gives

Thus any auction satisfying $\phi(Z_k)>C_k$ yields an upper bound on non-winner valuations,

whenever the numerator is positive. Conversely, any auction satisfying $\phi(Z_k)<C_k$ yields a lower bound on valuations,

whenever the numerator is positive.

The empirical content depends on how $C_k$ is specified. Here we outline our baseline specification, $C_k=\phi(Z_{k+1})$, which yields a valid bound under the following monotonicity condition on common values:

This condition is a transparent sufficient condition for deriving a recursive bound on $W_k(\theta_i)$. It should not be interpreted as a restriction on bidder preferences, market structure, or the auctioneer's ordering rule.\footnote{The earlier work of elmaghraby2003importance, kittsteiner2004declining, and shi2022implementing relates item ordering to revenue or efficiency under specific equilibrium models; our use of Condition (ref) is different in spirit, requiring no equilibrium model and no sorting rule.} When bidders view the full remaining sequence as their relevant opportunity set, Condition (ref) requires common values to be decreasing along that remaining sequence so that the value of waiting after period $k$ is bounded by the next opportunity. In practice, the condition can be imposed locally on any tail of the sequence: if $\phi(Z_k)>\phi(Z_{k+1})>\cdots>\phi(Z_K)$ for a given market, then the bounds below can be applied to the auctions in that decreasing tail even when earlier auctions in the same market do not satisfy the condition.

Part (a), which always holds under Condition (ref), provides an upper bound on $W_k(\theta_i)$ with $C_k=\phi(Z_{k+1})$ and $m_k=0$. Part (b) refines Part (a) by incorporating opening prices along the relevant future tail with $m_k=p_{k+1}+\Delta$.\footnote{Since $W_k(\theta_i) \geq 0$ (bidders can always abstain from future auctions), Part (b) is informative only for bidders whose relevant future net payoff is nonnegative. The condition $V_{i,t}\geq \max\{0,V_{i,t+1}\}$ is a sufficient recursive dominance condition: each future auction's maximum net payoff weakly exceeds both zero and the bound on later continuation payoffs. For adjacent periods with positive later net payoff, it requires $\theta_i\{\phi(Z_t)-\phi(Z_{t+1})\}\geq p_t-p_{t+1}$, so opening prices in both periods enter the comparison.} When the condition in Part (b) holds, combining Lemma (ref) with (ref) yields

Since $\phi(Z_k) > \phi(Z_{k+1})$ by Condition (ref), dividing by the positive quantity $\phi(Z_k) - \phi(Z_{k+1})$ yields an upper bound on $\theta_i$:

The first finite bound is called the baseline (gross) bound implied by Part (a) of Lemma (ref). The second is referred to as the sharper net bound implied by Part (b). When $k+2\leq K$, the first adjacent-period dominance condition is \[ \theta_i\phi(Z_{k+1})-(p_{k+1}+\Delta) \geq \max\{0,\theta_i\phi(Z_{k+2})-(p_{k+2}+\Delta)\}. \] When the later net payoff is positive, this is equivalent to $\theta_i\{\phi(Z_{k+1})-\phi(Z_{k+2})\}\geq p_{k+1}-p_{k+2}$. Thus, if opening prices are zero and the relevant future net payoffs are nonnegative, Condition (ref) makes the adjacent comparisons automatic; the condition is also automatic in the two-period case once the next auction is profitably enterable.

The upper bounds in (ref), (ref), and (ref) are most informative for the second-highest order statistic $\theta^{2:n_k}$. The bound on $\theta^{2:n_k}$ implies the same inequality on all lower-order statistics since $\theta^{j:n_k} \leq \theta^{2:n_k}$ for $j > 2$. The bound for the highest-order statistic is uninformative (only $\bar{\theta}$). When Lemma (ref)(b) applies, the relevant upper bounds are:

Bounds on the Valuation Distribution

We now translate the valuation bounds into bounds on the distribution $F^0$. Lemma (ref) implies the first-order stochastic dominance:

The distribution of the $j$th order statistic from $n_k$ i.i.d.\ draws from $F^0$ can be expressed via the incomplete beta function:

where $\Gamma(z; a, b) \equiv I_z(a,b) = \int_0^z \frac{(a+b-1)!}{(a-1)!(b-1)!} t^{a-1}(1-t)^{b-1}\, dt$ is the regularized incomplete beta function. Since $\Gamma^{-1}(\cdot)$ is monotone, inverting (ref) and combining with (ref) gives

The minimum across all periods and order statistics selects the tightest upper bound for each $x$.

For the lower bound on $F^0$, consider the last period. Define $\zeta_{1:n_K}^\Delta \equiv (b^{1:n_K} + \Delta)/\phi(Z_K)$ with distribution $G_{1:n_K}^\Delta(\cdot)$.\footnote{We use $\zeta$ rather than $\eta$ for this statistic to distinguish it from the reduced bid $\eta_{i,k} = b_{i,k}/\phi(Z_k)$; note that $\zeta_{1:n_K}^\Delta$ includes the bid increment $\Delta$ and uses the winning bid rather than a single bidder's bid.} The upper bound on $\theta^{2:n_K}$ from (ref) yields:

For periods $k < K$, define the baseline sequential statistic $h^g_{1:n_k} \equiv (b^{1:n_k}+\Delta)/(\phi(Z_k) - \phi(Z_{k+1}))$ with distribution $H^g_{1:n_k}(\cdot)$. Then:

Applying the beta inverse transformation and taking the maximum across all bounds:

for all $x \in [\underline{\theta}, \bar{\theta}]$. When Lemma (ref)(b) applies, the sharper net statistic $h^n_{1:n_k}\equiv (b^{1:n_k}-p_{k+1})/(\phi(Z_k)-\phi(Z_{k+1}))$ can replace $h^g_{1:n_k}$ in (ref)--(ref). When opening prices vary across auctions, the bounds $F_U$ and $F_L$ apply to the unconditional truncated distribution $F(\cdot)$ defined in (ref). Conditional on $\tilde{p}$, order statistics and valuations are i.i.d.\ from the truncated distribution; unconditional moments average the corresponding conditional order-statistic probabilities over the distribution of $\tilde{p}$.

When individual bidders can be tracked across periods in data, the panel structure provides an additional source of identifying information. For each bidder $i$ observed in periods $k \in \mathcal{K}_i$, Assumption (ref) implies $\eta_{i,k} = b_{i,k}/\phi(Z_k) \leq \theta_i$ for every period $k \in \mathcal{K}_i$. Taking the maximum across all observed periods:

Since $\eta_i^{\max} \leq \theta_i$ for every bidder $i$, the event $\{\theta_i \leq x\}$ implies $\{\eta_i^{\max} \leq x\}$. Therefore, with $G^{\max}(\cdot)$ denoting the distribution of $\eta_i^{\max}$:

This bound can be tighter than single-period beta-inversion bounds because it uses the maximum of a bidder's reduced bids across all auctions, not just one. Importantly, (ref) does not rely on the order statistics/beta function inversion, and does not require Condition (ref); it uses only Assumption (ref) applied across periods.

Robustness of the Bounds

The bounds can also accommodate several features commonly encountered in real auction markets as discussed below. Appendix (ref) further discusses no-bid periods and shows how they can be removed when the remaining sequence satisfies the relevant descending filter.

Sharp Bounds

The bounds we derived so far are not necessarily sharp. We now characterize the sharp bounds using the generalized instrumental variable (GIV) framework of chesher2017generalized. The value of this framework is that sharpness can be stated through random-set theory rather than via a constructive proof, which is difficult even in the static HT model. To simplify exposition, consider the fixed-$N$ case with the baseline specification $C_k=\phi(Z_{k+1})$ and $m_k=0$ (zero opening prices and $\Delta = 0$). Let $\mathcal{K}$ denote the terminal period together with the retained non-terminal periods satisfying the adjacent sign condition $\phi(z_k)>\phi(z_{k+1})$; periods that fail this filter are omitted from the sharp-set construction.\footnote{Other continuation-index specifications are handled by replacing $b_k^{1:N}/\{\phi(Z_k)-\phi(Z_{k+1})\}$ below with the statistic in (ref).} For each $k\in\mathcal{K}$, define the vector of ordered bids $B_k \equiv (b_k^{1:N}, \ldots, b_k^{N:N})$ and item characteristics $Z_k$. Let $U \equiv (u_1, \ldots, u_N)$ denote the vector of transformed value order statistics, $u_j \equiv F^0(\theta^{j:N})$, so that $U$ is supported on $\mathcal{R}_U=\{1\geq u_1\geq\cdots\geq u_N\geq0\}$ with density $N!$.

Define $B \equiv (B_k)_{k\in\mathcal{K}}$ and $Z \equiv (Z_k)_{k\in\mathcal{K}}$. The inequalities (ref) and (ref) imply the structural function $h(B,Z,U)=\sum_{k\in\mathcal{K}} h_k(B_k,Z_k,U)=0$ a.s., where

with $\phi(Z_{K+1})\equiv0$. For a closed subset $S\subseteq\mathcal{R}_U$, let $G_U(S)\equiv P[U\in S]$. Given observables $B=b$ and $Z=z$, the set of latent variables consistent with the model is

When $B$ is random, $\mathcal{U}(B,z;h)$ is a random set. Let $\mathcal{Q}(z;h)\equiv\{\mathcal{U}(b,z;h): b\in\mathcal{R}_B\}$ be its conditional support and $\mathcal{Q}^*(z;h)$ denote the collection of unions of elements of $\mathcal{Q}(z;h)$.

The sharp set therefore consists of all valuation distributions whose induced structural function satisfies an uncountable collection of Artstein inequalities. The bounds in (ref) and (ref) correspond to particular contiguous unions of $U$-level sets. To see this, write reduced bids as $\eta_k=(\eta_{k,1},\ldots,\eta_{k,N})$ with $b_k(\eta_k)=(\eta_{k,1}\phi(z_k),\ldots,\eta_{k,N}\phi(z_k))$ and $\eta_{k,1}\geq\cdots\geq\eta_{k,N}$, and define $b(\eta)\equiv(b_k(\eta_k))_{k\in\mathcal{K}}$. For two arrays $\eta'$ and $\eta''$, define

where $[\eta',\eta'']$ is the Cartesian product of componentwise intervals. For the class of intervals used below, this union occupies

Applying Lemma (ref) to these contiguous unions yields moment inequalities that every admissible $F^0$ must satisfy.

The no-overbidding upper bound is recovered by fixing a period $k$ and rank $m$, setting $\eta_{k,1}''=\bar{\theta}$, $\eta_{k,j}'=\underline{\theta}$ for $j>m$, and $\eta_{k,j}'=\theta$ for $j\leq m$, while leaving the restrictions for other periods uninformative. The resulting inequality is

where $G_{k,m:N}$ is the distribution of the reduced-bid order statistic $\eta_k^{m:N}$. Since $\Gamma(\cdot;N-m+1,m)$ is increasing, \[ F^0(\theta) \leq \Gamma^{-1}\!\big(G_{k,m:N}(\theta);\, N-m+1,\, m\big), \quad \forall k\leq K, \] which gives the upper bound in (ref) after taking the minimum over periods and order statistics.

The lower bounds are recovered analogously. For the terminal period, set $\eta_{K,1}''=\theta$ and leave all other restrictions uninformative. Then

For a non-terminal period $k<K$, set $\eta_{k,1}''=\theta\{\phi(z_k)-\phi(z_{k+1})\}/\phi(z_k)$ and again leave other restrictions uninformative. This gives

where $H_{k,1:N}$ is the distribution of $\eta_k^{1:N}\phi(z_k)/\{\phi(z_k)-\phi(z_{k+1})\}$. Inverting (ref) and (ref) gives the lower bound in (ref).

These examples show that (ref) and (ref) are necessary implications of the sharp GIV characterization, but they do not exhaust it. Additional inequalities combine restrictions across order statistics and periods. For instance, for $a,b\in[\underline{\theta},\bar{\theta}]$ with $b\leq a$ and $a\phi(z_k)/\{\phi(z_k)-\phi(z_{k+1})\}\in[\underline{\theta},\bar{\theta}]$,

This inequality restricts the increase in $F^0$ between $b$ and $a\phi(z_k)/\{\phi(z_k)-\phi(z_{k+1})\}$. Since the sharp characterization involves an uncountable collection of such inequalities, implementation is computationally demanding. The empirical analysis therefore uses the tractable non-sharp bounds, while the GIV characterization clarifies exactly where additional identifying content resides.

Estimation and Inference

This section addresses the statistical challenges of estimating the bounds from finite samples and constructing confidence intervals for the partially identified distribution $F^0$.

Estimation when $N$ is fixed

When each auction has the same number of bidders $N$, the distribution bounds in (ref) and (ref) are computed by inverting the incomplete beta function $\Gamma^{-1}(\hat{G}_{j:N}(x);\, a,\, b)$, where $\hat{G}_{j:N}$ is the empirical CDF of the observed $j$th-order statistic. This inversion is well-behaved in the population but exhibits two fundamental difficulties in finite samples. First, the mapping from order-statistic distributions to the parent distribution is not Lipschitz continuous at the support boundaries. menzel2013large show that the derivative of $\Gamma^{-1}$ is unbounded at the endpoints, so that small estimation errors in $\hat{G}$ are amplified without bound near the tails. The resulting optimal convergence rate for nonparametric estimation of $F^0$ from the $j$th-order statistic is as slow as $T^{-1/N}$, where $T$ is the number of auctions, dramatically slower than the usual $T^{-2/5}$ rate for density estimation. cherapanamjeri2022estimation further establish that full-support recovery of $F^0$ in Kolmogorov distance is impossible from order statistics, with the exponential dependence on $N$ constituting a fundamental lower bound that no estimator can improve upon. Second, the empirical CDF $\hat{G}_{j:N}$ is degenerate in the left tail for high order statistics. Since $\Pr(\theta_{j:N} \leq v) = \Gamma(F^0(v);\, N\!-\!j\!+\!1,\, j) \approx [F^0(v)]^{N-j+1}$, which vanishes rapidly for small $F^0(v)$, the ECDF of the $j$th-order statistic has zero mass below a cutoff that grows with $N-j+1$. Inverting $\hat{G}_{j:N}(v) = 0$ gives $\Gamma^{-1}(0) = 0$, collapsing the bound to zero. This creates a “dead zone” in the left tail of $F_U$ that widens with the number of bidders, precisely the region where the bounds should be widest. These limitations are shared by all nonparametric bounds approaches based on order statistic inversion, including the HT framework.

Equation (ref) defines the population upper bound as the minimum of $\Gamma^{-1}(G_{j:N}(x);\, N\!-\!j\!+\!1,\, j)$ across all order statistics $j = 1, \ldots, N$. Each order statistic provides the strongest identification power at different quantiles: the second-order statistic ($j = 2$) is most informative in the upper tail but degenerate in the lower tail, while the minimum ($j = N$) is informative in the lower tail but quickly becomes degenerate as one approaches the right tail. In the population, the intersection (minimum) across all $j$ yields the sharpest bound. In finite samples, however, the na\"ive sample analogue (taking the pointwise minimum of $\Gamma^{-1}(\hat{G}_{j:N})$ across $j$) inherits the worst finite-sample pathology of each $j$, since the minimum of noisy estimators is biased downward chernozhukov2013intersection. The original HT estimator addresses finite-sample extremum bias by replacing the pointwise min/max across order-statistic inequalities with a smooth exponential-weighted approximation. Rather than implement this ad hoc smoothing device, we include the bias-corrected intersection estimator of chernozhukov2013intersection, which provides a systematic modern alternative for the same intersection-bounds problem. We consider the following six alternative approaches to address this problem:

We verify these approaches' finite-sample performance in Monte Carlo simulations in Section (ref).

Estimation when $N$ varies across auctions

When the number of bidders $N_a$ varies across auctions, as is typical in practice, the beta inversion (ref) cannot be applied directly with a common $N$. A natural estimator pools all auctions while respecting each auction's individual $N_a$. For the $j$th order statistic, the no-overbidding inequality implies the population moment \[ \mathbf{E}\!\left[\mathbf{1}\{\eta^{j:N_a}\leq v\}\right] \geq \mathbf{E}\!\left[I_{F^0(v)}(N_a-j+1,j)\right], \] where $I_x(a,b) \equiv \Gamma(x; a, b)$ is the regularized incomplete beta function defined in (ref). The estimator replaces expectations by sample analogs and solves

The upper bound $F_U(v)$ is the solution to the equation obtained by replacing the inequality with equality. Since $I_F(a,b)$ is monotone increasing in $F$ for each $(a,b)$, the right-hand side is monotone in $F$ and the equation can be solved by bisection at each grid point $v$.

The lower bound has an analogous moment condition. By Theorem (ref), the baseline sequential statistic $h^{g,a}_{1:N_a} = (b^{1:N_a}_a+\Delta)/(\phi(Z_{k,a}) - \phi(Z_{k+1,a}))$ bounds every non-winner valuation from above; since only the winner is unrestricted, it follows that $h^{g,a}_{1:N_a}\geq\theta^{2:N_a}$. Hence \[ \mathbf{E}\!\left[\mathbf{1}\{h^g_{1:N_a}\leq v\}\right] \leq \mathbf{E}\!\left[I_{F^0(v)}(N_a-1,2)\right]. \] The sample analog solves

The lower bound $F_L(v)$ is the solution to the equation obtained by replacing the inequality with equality; since the right-hand side is strictly monotone increasing in $F$, the inequality direction is preserved and $F_L(v) \leq F^0(v)$. If Lemma (ref)(b) applies, the same moment condition can be evaluated with the sharper net statistic from (ref).

For the upper bound, the no-overbidding inequality $\eta^{j:N_a} \leq \theta^{j:N_a}$ holds for every $j \in \{1, \ldots, N_a\}$, so any rank could in principle be used in (ref) with the corresponding beta shape $(N_a\!-\!j\!+\!1, j)$. We restrict to $j = 2$ in our implementation for three reasons. First, $j = 1$ is essentially uninformative: the regularized incomplete beta $I_F(N_a, 1) = F^{N_a}$ is close to zero except for $F$ very near one, so inverting the winning-bid ECDF yields $F_U(v) \approx 1$ across the bulk of the support. Second, ranks $j \geq 3$ require beta shapes $(N_a - j + 1, j)$ that differ from the shape pinned down by the lower bound, $(N_a\!-\!1, 2)$. Combining bounds across mismatched shapes can produce $F_U < F_L$ in finite samples and complicates the construction of a coherent confidence interval; using $j = 2$ matches the lower bound's beta shape and helps preserve coherence. Finite-sample crossings can still occur when endpoints are computed from different samples or multiple moments are combined. Third, with heterogeneous $N_a$, the variation in $N_a$ already plays the role that variation in $j$ would play under fixed $N$: low-$N_a$ auctions have rapidly-rising beta weights $I_F(N_a\!-\!1, 2)$ in the left tail, supplying the left-tail identification that the adaptive multi-$j$ method of Section (ref) is designed to recover. The adaptive method is therefore unnecessary in our empirical applications, and we use a single $j = 2$ throughout.

This moment-condition inversion follows the general approach to estimation under partial identification with moment inequalities chernozhukov2007estimation, adapted to the specific structure of order statistic bounds with heterogeneous $N_a$. In the original application of haile2003inference, bounds are estimated by conditioning on $N$ via kernel smoothing, restricting the estimation window to auctions with similar numbers of bidders; as HT note, allowing too much heterogeneity in $N$ within the bandwidth causes the bounds to cross. Our approach takes a different strategy: rather than conditioning on $N$, it pools all auctions and lets the beta function weights $I_F(N_a\!-\!1, 2)$ absorb the heterogeneity in $N_a$ directly. This avoids both the information loss from narrow-bandwidth conditioning and the crossing problem from wide-bandwidth smoothing.

The approach has two advantages over the fixed-$N$ approach. First, it uses all $T$ auctions simultaneously with their individual $N_a$, avoiding the information loss from partitioning into per-$N$ subsamples. Second, heterogeneity in $N_a$ provides identification across the full support of $F^0$ through a natural weighting mechanism. Each auction $a$ contributes a beta CDF $I_{F(v)}(N_a\!-\!1, 2)$ to the right-hand side of (ref). The shape of this beta CDF depends on $N_a$: for a small-$N$ auction (e.g., $N_a = 3$), $I_F(2, 2) = 3F^2 - 2F^3$ rises at much smaller values of $F$ than high-$N$ beta CDFs, contributing information in the left tail of the distribution. For a large-$N$ auction (e.g., $N_a = 30$), $I_F(29, 2) = 30F^{29} - 29F^{30}$ is essentially zero until $F$ is close to 1, contributing information only in the right tail. The bisection algorithm finds the single $F(v)$ that makes the average of these heterogeneous beta CDFs equal the observed empirical CDF. In effect, low-$N$ auctions receive higher “weight” in the left tail and high-$N$ auctions receive higher weight in the right tail, with the weighting determined endogenously by the beta function shapes. The variation in $N_a$ thus plays the same role as variation in the order statistic rank $j$ in the adaptive method, providing a natural alternative to combining multiple order statistics.

The following proposition establishes validity, consistency, and asymptotic normality of the moment-condition inversion estimator.

Inference

The valuation distribution $F^0$ is partially identified, lying between the population bounds $[F_L(v),F_U(v)]$ for each $v$. Standard pointwise confidence intervals around either estimated bound are insufficient because they account only for sampling error in that endpoint, not for the identification uncertainty represented by the width of the bounds. We construct confidence intervals using two complementary approaches. The first approach is the nonparametric bootstrap. We resample auctions with replacement $B$ times, recomputing the bounds for each bootstrap sample.\footnote{When upper- and lower-bound statistics are computed from distinct pooled samples, each sample can be resampled using its own observation unit. When the same auction or market can contribute to both endpoints, a paired or clustered bootstrap at the auction/market level preserves the covariance between the two endpoints.} The pointwise $(1-\alpha)$ bootstrap interval uses Bonferroni endpoint quantiles: \[ \hat{F}_U^{\text{conf}}(v) = Q_{1-\alpha/2}\!\left(\hat{F}_U^{*b}(v);\, b = 1,\ldots,B\right), \] and $\hat{F}_L^{\text{conf}}(v) = Q_{\alpha/2}(\hat{F}_L^{*b}(v))$ for the lower bound. The confidence bounds $[\hat{F}_L^{\text{conf}}, \hat{F}_U^{\text{conf}}]$ are wider than the estimated bounds, and $\Pr(\hat{F}_L^{\text{conf}}(v) \leq F^0(v) \leq \hat{F}_U^{\text{conf}}(v)) \geq 1 - \alpha$ at each $v$.

imbens2004confidence propose confidence intervals for a parameter $\theta \in [\theta_L, \theta_U]$ that cover the true $\theta$ (not the identified set) with asymptotic probability at least $1-\alpha$. The CI is \[ \text{CI}_\alpha = \left[\hat{F}_L(v) - c_n \cdot \hat{\sigma}_L(v),\;\; \hat{F}_U(v) + c_n \cdot \hat{\sigma}_U(v)\right], \] where $\hat{\sigma}_L$ and $\hat{\sigma}_U$ are standard errors of the bound estimators, computed analytically via (ref), and $c_n$ solves $\Phi(c_n + \hat{\Delta}_n / \max(\hat{\sigma}_L, \hat{\sigma}_U)) - \Phi(-c_n) = 1 - \alpha$, with $\hat{\Delta}_n = \max(\hat{F}_U - \hat{F}_L, 0)$. When the bounds are wide, $c_n \to z_{1-\alpha}$ (one-sided on each end); when they collapse to a point, $c_n \to z_{1-\alpha/2}$ (two-sided).

For the moment-condition estimator (ref), the standard errors $\hat\sigma_U$ and $\hat\sigma_L$ can be computed analytically via the delta method applied to the implicit equation $m(\hat F, v) = 0$. Let $Y_a$ denote the statistic entering the relevant endpoint, and let $(s_{1a},s_2)$ denote the corresponding beta shape: $(N_a-j+1,j)$ for the upper bound and $(N_a-1,2)$ for the lower bound. With $m(F,v)\equiv T^{-1}\sum_a[I_F(s_{1a},s_2)-\mathbf{1}\{Y_a\leq v\}]$, the implicit function theorem gives

where $f_{\text{Beta}}(\cdot; a, b)$ is the beta density. The numerator is the standard error of the moment evaluated at the estimated bound; the denominator converts moment uncertainty into $F$-space via the average beta density. When $N_a$ is fixed, (ref) reduces to the familiar expression $\sqrt{\hat G(v)(1-\hat G(v))/T}/f_{\text{Beta}}(\hat F(v);s_1,s_2)$. We use the analytical formula for the reported Imbens--Manski bands because it avoids bootstrap noise and produces smooth confidence bands, while treating the bootstrap as the more conservative finite-sample benchmark. Both approaches provide pointwise coverage at each $v$. For uniform coverage across the entire support, one could calibrate via the sup-$t$ bootstrap chernozhukov2007estimation, at the cost of wider bands.

Simulation

We evaluate the finite-sample performance of the six estimation approaches with fixed $N$, the moment-condition inversion method with varying $N_a$, and the inference procedures from the previous section. We also compare the proposed lower bounds to three alternatives: (i) the static bounds of haile2003inference applied naively to each period (HT); (ii) the last-period-only bounds, which discard all non-terminal auctions; and (iii) the bidder-level panel bounds from Section (ref).

Design and Bound Comparison with fixed $N$

We generate bid data from a structural two-period sequential English auction model. In each simulation round, private values $\theta_i$ for $N = 10$ bidders are drawn i.i.d.\ from $\text{LogNormal}(\mu = 1, \sigma = 0.5)$. Common values are drawn independently across auctions. Bids are generated from the symmetric Bayesian Nash equilibrium of the sequential second-price auction kittsteiner2004declining. In the second (last) period, each remaining bidder's maximum willingness to pay is $\text{MWP}_{i,2} = \phi(Z_2)\theta_i$. In the first period, forward-looking bidders account for the option value of winning in the future:

where the integral represents the expected continuation value, the expected gain from participating in the second auction conditional on being the strongest remaining bidder. The lower integration limit accounts for the reserve price: bidders with $\theta_i < p_2/\phi(Z_2)$ cannot profitably enter the second auction and have zero continuation value.

We simulate auctions starting at an opening price $p_k$. Bidders drop out when the price reaches their MWP. To introduce departures from equilibrium behavior, bidders ranked third or lower (by MWP) drop out at a random fraction $\omega_i \sim \text{Uniform}[0.3, 1]$ of their MWP, creating substantial noise in observed lower-ranked bids. The second-highest bidder drops at their full MWP, and the winner bids one increment $\Delta = 0.05$ above the second-highest dropout price. We consider three configurations. In all scenarios, the opening price in period 2 is stochastic: $p_2 = \phi(Z_2) \cdot U(0.5, 1)$, so that the reserve price scales with the item's common value.

For each scenario, we run 500 Monte Carlo replications; in each replication we generate 500 sequential auction pairs, compute all bounds using the full set of observed bids, and average the bounds and coverage statistics across replications. We first examine the performance of the lower bounds, where our approach differs most from the alternatives. Figure (ref) displays the mean bounds for the three scenarios. In all three scenarios, the HT lower bound (in the middle row of Figure (ref)) lies above the no-overbidding upper bound over a substantial portion of the support. This occurs because HT treats first-period bids as if they were stand-alone valuations, producing inflated lower bound statistics. By contrast, the proposed upper and lower bounds (in the top row of Figure (ref)) bracket the true $F^0$ across the bulk of the support in the large-drop scenario. In the small-drop and supply-uncertainty scenarios, the bounds are extremely narrow, so that even minor finite-sample noise can cause the bounds to cross at isolated grid points. The stochastic opening price $p_2 = \phi(Z_2) \cdot U(0.5, 1)$ plays a key role in identification. In the sequential bound $\theta_i \leq (b^{1:n_1} - p_2)/(\phi_1 - \phi_2)$, a positive $p_2$ reduces the numerator, tightening the upper bound on valuations and hence the lower bound on $F$. In the simulations using this sharper net statistic, we retain observations satisfying $b^{1:n_1}/\phi_1 > p_2/\phi_2$. Simultaneously, the positive reserve screens out low-value bidders from period 2, weakening the last-period bound. The sequential structure therefore provides the greatest identification power when opening prices are non-trivial.

We now compare the six approaches from Section (ref) for combining multiple order statistics in the upper bound $F_U$. All methods share the same lower bound; only the upper bound computation differs. We use the large-drop scenario ($\Delta = 0.05$, dropout $\omega_i \sim U(0.3, 1)$, $p_2 = \phi(Z_2) \cdot U(0.5, 1)$, 2{,}000 auctions across 200 replications) and vary the number of bidders: $N \in \{5, 10, 20\}$. The theory in Section (ref) predicts that the left-tail problem should worsen rapidly with $N$. Table (ref) reports the results. We measure coverage (fraction of $v$ where $F_L(v) \leq F^0(v) \leq F_U(v)$), left-tail coverage (coverage restricted to $v$ where $F^0(v) < 0.1$), and invalidity (fraction of $v$ where $F_U(v) < F^0(v)$).

The left-tail problem worsens dramatically with $N$: for the baseline method, left-tail coverage falls from the low single digits at $N = 5$ to near zero at $N = 20$, and the “dead zone” (the smallest $v$ at which $F_U(v) > 0$) approximately doubles with each doubling of $N$, consistent with the $T^{-1/N}$ rate in menzel2013large. By contrast, the adaptive $j$-selection method dominates the other methods. By excluding order statistics with degenerate empirical CDFs and using lower-ranked order statistics (which have support in the left tail), it achieves 100% left-tail coverage at $N \geq 10$ with substantially lower invalidity than the baseline. The adaptive method also has the best overall coverage among valid methods at every $N$. The CLR bias correction and Bernstein smoothing provide minimal improvement, or even degrade performance, relative to the baseline. The CLR correction addresses extremum bias (the selection effect from taking the minimum), but this bias is negligible relative to the structural problem: the $j = 2$ empirical CDF is literally zero in the left tail, and no bias correction can recover information that is not in the data. The Bernstein smoother pushes the empirical CDF positive near the boundary but overshoots, producing overly aggressive bounds with high invalidity. IVW shows left-tail coverage similar to the adaptive approach, but worse overall coverage and higher invalidity. The shape of the adaptive upper bound is displayed in the bottom panel of Figure (ref).

Having established that the adaptive method dominates, we now examine whether proper inference preserves this advantage. Table (ref) reports the frequentist coverage of the 95% confidence intervals described in Section (ref), based on 200 Monte Carlo replications with 200 bootstrap draws each, comparing the baseline ($j = 2$ only) and adaptive approaches. The confidence bands substantially improve coverage: moving from estimated bounds to 95% CIs raises coverage for the adaptive method by roughly 20 percentage points across all $N$. At $N = 20$, the adaptive method with both bootstrap and Imbens--Manski CIs achieves coverage at or above the nominal 95% level with near-zero invalidity, demonstrating that the combination of adaptive $j$-selection and proper inference yields reliable finite-sample bounds even in high-competition settings where the baseline approach fails entirely in the tails.

Heterogeneous Number of Bidders

In practice (as in both of our empirical applications), the number of bidders tends to vary across auctions. We validate the proposed moment-condition inversion approach by simulating 500 sequential auctions across 200 replications with $N_a \sim \text{DiscreteUniform}(2, 20)$, using the large-drop scenario. Table (ref) compares three methods for the bound estimates and reports 95% CI coverage. The moment-condition inversion produces the tightest bounds (width 0.086, invalidity 0.8%), outperforming the fixed median-$N$ proxy (width 0.097, invalidity 1.7%). The fixed-$N =$ min approach is essentially useless: using $N = 2$ for all auctions produces a bound with 87% invalidity, because the beta shape $(1, 2)$ vastly overestimates the identifying power of high-$N_a$ auctions. With $N_a \sim \text{DiscreteUniform}(2, 20)$, the 95% confidence intervals achieve 99.7% overall coverage and 100% left-tail coverage. The strong performance reflects the presence of low-$N_a$ auctions in the sample: auctions with $N_a = 2$ or $3$ contribute informative beta weights at small $F$, populating the left tail where the $j = 2$ ECDF would otherwise be degenerate. This confirms that heterogeneity in $N_a$ provides a natural solution to the left-tail dead zone problem when low-$N$ auctions are present in the data.

Figure (ref) displays the mean bounds across 200 Monte Carlo iterations. The moment-condition inversion and fixed-median bounds are visually close, while the fixed-min approach produces bounds that are too tight and lie below the true distribution over much of the support. In the figure, upper-bound curves are dashed and lower-bound curves are solid.

Empirical Applications

Korean Wholesale Used Car Auctions

We use data from a wholesale used car auction house located in Suwon, South Korea, which opened in May 2000 as the first fully computerized automobile auction facility in the country il2014nonparametric, roberts2013unobserved. The auction house held weekly sales from October 2001 through December 2002, our sample period. Between 250 and 380 registered dealer members were eligible to bid, with roughly half attending any given weekly auction. The pool of registered dealers remained stable throughout the sample period. Sellers, predominantly individual car owners, rental companies, and corporate fleet operators, consigned vehicles and paid a listing fee of approximately \$50. The auction house charged a 2.2% commission on the selling price to both buyer and seller upon sale. A few days before each weekly auction, potential buyers received a catalog with detailed information about every car to be sold, including make, model, year, mileage, engine size, transmission type, fuel type, and accident history. Buyers could also physically inspect the cars 2--3 hours before the auction.

On each auction day, several hundred to over a thousand cars were offered sequentially (median 733 consignments per day) to a stable pool of 150--200 active dealers. Approximately half of consigned cars sold at any given auction and unsold cars were typically reconsigned for a future auction day. The auction followed an open ascending-price format similar to the English button auction milgrom1982theory. Each car had a reserve price set by the seller. The auctioneer announced the opening price and the price rose in fixed increments of 30{,}000 KRW (approximately \$20) every 3 seconds. Buyers signaled willingness to buy at the current price by pressing a button on an electronic device. The auction house displayed only the current price and a traffic-light indicator: green ($\geq 3$ active bidders), yellow (2 active bidders), and red (1 active bidder). Bidder identities were not revealed. Importantly, reentry was permitted: a buyer could drop out temporarily and rejoin later if the price remained competitive, distinguishing this format from the strict button auction where exit is irrevocable. When only one buyer remained active, the auction ended and the winner paid the current price. A car that sold was typically paid for and removed within a few days.

The raw data contain 160 auction days, each comprising a series of sequential auctions. We observe the top three bids and corresponding bidder identifiers for each auction, along with vehicle characteristics and auction outcomes. Of 100{,}858 total auction records, 53{,}241 resulted in a sale, of which 48{,}753 have the top-two bids recorded and enter our analysis. To maintain the IPV assumption, we estimate bounds separately within five major vehicle segments: automatic sedans (gasoline), manual sedans (gasoline), manual non-sedans (gasoline), automatic non-sedans (gasoline), and manual non-sedans (diesel). These five categories cover 43{,}165 of the 48{,}753 usable sold auctions (88.5%). Within each category, we construct all consecutive auction pairs.

Estimation proceeds in two steps. In non-terminal periods, bids are strategically discounted by the continuation value: $b_{i,k} = \theta_i\phi(Z_k) - \text{CV}_k(\theta_i)$. This breaks the multiplicative structure needed for hedonic identification, since $\log b_{i,k} \neq \log\theta_i + \log\phi(Z_k)$. Only in the last period, where $\text{CV}_K = 0$, does the log-additive decomposition hold exactly. We therefore use last-period bids to estimate $\phi(Z_k)$ via hedonic regression:

where the superscript $m$ indexes the sequential auction series and $Z_K^m$ collects observed characteristics of the item in the last period of series $m$. Bidder fixed effects absorb $\log\theta_i$, and the coefficients on car characteristics identify $\phi(\cdot)$. Standard errors are clustered at the auction-day level. Taking $\hat{\phi}(Z_k)$ as given, we estimate the upper and lower bounds on $F(\cdot)$. Table (ref) reports the hedonic regression estimates. The coefficients have expected signs: older cars sell for less, larger engine size and higher condition ratings increase value, and the point estimate for imported cars is positive.

The total number of potential bidders is not directly observed. We observe only the top three bids per auction, while the pool of registered dealers may not all be active in every market. We proxy for $N_a$ in each market by counting the number of unique dealers who won at least one auction within that market. The estimated $N_a$ varies substantially: the median ranges from 28 (manual non-sedan diesel and automatic non-sedan gasoline) to 57 (automatic sedan gasoline), reflecting different competition intensities and popularities across vehicle segments. We use the moment-condition inversion method to estimate the bounds because the data exhibit heterogeneous $N_a$. Figure (ref) displays the estimated upper bound and three lower-bound specifications. The baseline lower bound uses the adjacent-opportunity specification, with $C_k=\hat\phi(Z_{k+1})$ and $m_k=0$, combined with the terminal-period inequality in Lemma (ref) without imposing the global monotonicity condition.\footnote{Under this specification, 15{,}817 sequential pairs that satisfy the filter $\hat\phi(Z_k)>\hat\phi(Z_{k+1})$ are retained to construct the lower bound.} This specification treats the next car in the same auction-day category as the relevant substitute opportunity in the dealer's marginal procurement problem. The interpretation is natural in the Korean used-car auction setting because vehicles are sold in a posted sequence, dealers observe a dense same-day stream of similar alternatives, and the immediately following auction provides a transparent proxy for the continuation opportunity. Setting $m_k=0$ bounds continuation value by the gross value of the next opportunity without using the opening price. Thus all descending adjacent pairs with positive finite statistics contribute. The baseline bounds have mean widths of 0.029--0.107 across categories.

Figure (ref) also reports the sharper net specification (subtracting the next-period opening price), retaining pairs that satisfy the normalized positivity filter $b^{1:n_k}/\hat\phi(Z_k)>p_{k+1}/\hat\phi(Z_{k+1})$, as well as the terminal-only lower bound. The sharper net bounds have mean widths of 0.029--0.111. They are not uniformly tighter because changing the statistic also changes the retained sample and the moment-condition inversion weights. Appendix (ref) conducts robustness checks that instead use the conservative full-future index $C_k=\max_{t\geq k+1}\hat\phi(Z_t)$ and tail-to-terminal subsequences that satisfy Condition (ref). The resulting loss of identifying power is concentrated in the left tail, while the bounds are largely unchanged over the rest of the support. The 95% Imbens--Manski confidence intervals have mean widths of 0.048--0.136.

Dealers in this market are wholesale buyers who often purchase multiple cars per auction day. We interpret the unit-demand assumption at the level of a marginal procurement opportunity rather than at the level of the dealer's entire daily business. A dealer bidding on a given vehicle decides whether to satisfy a current inventory need now or preserve the option to purchase a similar vehicle later, possibly at a lower price. If the dealer wins, that marginal need is satisfied; subsequent participation can be interpreted as a new procurement spell, or equivalently as a new bidder in the model. Under this renewal interpretation, observed multi-car purchases by the same dealer are consistent with unit demand within each decision problem. As a robustness check for the more literal fixed multi-unit-demand interpretation, we compute the terminal-only lower bound. This bound requires no sequential opportunity-cost restriction and is therefore valid even if dealers have several independent purchase needs within the same auction day. The resulting terminal-only bounds have mean widths of 0.029--0.129 and are largely consistent with the sequential bounds, though wider in the lower tail.

Cars and Bids Online Auctions

We apply our bounds approach to a second dataset: online ascending auctions from the Cars and Bids platform. This setting differs fundamentally from the Korean data in one key respect: there is no terminal auction. Individual listings run for seven days, new auctions open continuously, and a bidder tracking a particular vehicle segment faces an ongoing stream of opportunities with no defined endpoint. In this environment, the static HT framework cannot produce a valid lower bound on $F$ because waiting for a future listing is always a rational alternative. Our sequential approach fills this gap by exploiting the opportunity cost structure across consecutive auctions to provide a lower bound. The Cars and Bids data also offer two practical advantages: (i) the number of unique bidders $N$ is known exactly from complete bid histories, and (ii) the moderate-competition environment tests the finite-sample methods developed in Section (ref).

Cars and Bids is a U.S.-based online auction platform specializing in enthusiast vehicles. We use data from January 2024 through July 2025, comprising 10{,}811 auctions after filtering to auctions with at least two bidders and valid geographic information (mapped to U.S. Metropolitan Statistical Areas). Unlike the Korean data, complete bid histories are observed: every bid amount, bidder identity, and timestamp. The median auction attracts about 14 unique bidders (range 2--44), each placing approximately two bids on average, with a median winning bid of \$18{,}750 (mean \$29{,}631). We group vehicles into five market segments using $k$-prototypes clustering, a mixed-data analogue of $k$-means, on year, make, model, body style, and transmission type, imposing the IPV assumption within clusters rather than across the full dataset.

We construct bidder-specific sequences from the complete bid histories. For each bidder, we first record the auctions in which the bidder placed at least one bid, merge those bidder-auction observations with the auction's MSA, end month, end date, and estimated common value, and then order the auctions in which the bidder participated within each MSA-month cell. Consecutive observations in this bidder-specific ordering define a potential current and future opportunity pair. We retain pairs for which the current auction has a higher estimated common value than the bidder's next observed auction. This yields 4{,}137 sequential pairs from 1{,}388 unique bidders, roughly half of all 8{,}489 consecutive pairs.

We estimate the common value function $\phi(Z_k)$ via hedonic regression of $\log(\text{highest bid})$ on vehicle characteristics, using 10{,}811 auctions with make and model fixed effects. The car-characteristics model includes standard attributes such as year, mileage, engine, transmission, drivetrain, and body style. We use this car-characteristics-only model to construct $\hat\phi(Z)$ because $\phi(Z)$ is intended to capture predetermined object characteristics. For comparison, we also estimate a full model that adds text-based features from the listing: the length of seller highlights, service history, modifications, and seller notes, as well as the number of images and videos. We also incorporate comment sentiment features derived from a lexicon-based analysis of auction comments.\footnote{We use the AFINN sentiment lexicon nielsen2011new, which assigns integer scores from $-5$ (most negative) to $+5$ (most positive) to approximately 2{,}500 common English words (e.g., “excellent” $= +3$, “broken” $= -3$). For each auction, we tokenize all user comments posted on the listing page into individual words, match them against the AFINN lexicon, and compute the weighted sentiment as the average AFINN score across all matched words. This score measures the overall tone of community engagement: a high value indicates enthusiastic, positive commentary, while a low value suggests concerns or criticism. We also include the count of matched sentiment-bearing words as a separate regressor, capturing the volume of substantive discussion.} These variables improve price prediction but may partly reflect endogenous attention, information revealed during the auction, or bidder sentiment, so they are not used in the baseline common-value index. Table (ref) reports key coefficients from the hedonic regression. The car-characteristics-only model used for $\hat\phi$ achieves an adjusted $R^2$ of 0.819; including text and sentiment features raises it to 0.845.

In the Korean application, the hedonic regression uses only last-period bids and includes bidder fixed effects, because the continuation value $\text{CV}_k(\theta_i)$ in non-terminal periods breaks the log-additive structure needed for identification. In the Cars and Bids setting, this approach is infeasible for two reasons. First, each Cars and Bids listing is an independent 7-day online auction; the sequential structure is imposed ex post by tracking individual bidders across separate auctions, so there is no platform-defined terminal period that applies to all bidders simultaneously. Second, unlike the stable pool of 380 registered dealers in the Korean data, Cars and Bids bidders are highly heterogeneous and many participate in only one or two auctions, making bidder fixed effects impractical. Instead, we estimate the common-value index $\phi(Z)$ from the auction-level transaction price in the full cross-section of Cars and Bids auctions. The transaction price is the most informative object-level price signal available in the absence of terminal auctions or a stable bidder panel. Sequential incentives may affect transaction prices through bidder-specific continuation values; the maintained requirement is that, after conditioning on rich vehicle characteristics and make/model fixed effects, these residual strategic components are not systematically related to the observed characteristics used to construct the common-value index. The sequential structure is then accounted for in the second step through the opportunity-cost bounds, not by treating the auction as static.

With complete bid histories, we observe the exact number of unique bidders $N_a$ in each auction $a$. We use the moment-condition inversion method with each auction's exact $N_a$. For the lower bound, we implement the myopic continuation specification $C_k=\hat\phi(Z_{k+1})$ with zero opening prices. Because the Cars and Bids market is effectively infinite horizon, bidders only know the listings that are currently active but not the future composition of cars that will arrive on the platform. We therefore treat the bidder's next observed auction in the local sequence as the relevant continuation option. The resulting statistic is $h_{1:n_k} = b^{1:n_k}/(\hat\phi(Z_k) - \hat\phi(Z_{k+1}))$. We use $j = 2$ for the upper bound on $F$ (no-overbidding). Table (ref) reports the bound width for each vehicle cluster. The bounds have mean widths of 11.7--18.1 percentage points across clusters. The 95% Imbens--Manski confidence intervals are 15--21 percentage points wide, properly accounting for estimation uncertainty.

Since complete bid histories are available, we can also construct a panel upper bound from individual bidders' maximum reduced bids: $F(v) \leq G^{\max}(v) = \widehat{\Pr}(\eta_i^{\max} \leq v)$, using 22{,}000--49{,}000 unique bidders per cluster. This bound requires no beta inversion and no assumption on $N_a$. Comparing the panel upper bound to the sequential lower bound reveals that the baseline lower bound (computed with actual $N_a$) exceeds the panel upper bound at 20--30 grid points per cluster. Under the maintained assumptions, both bounds should be valid, so this crossing suggests that not all $N_a$ bidders are effectively competing in the auction. Casual bidders who place low bids may have systematically lower valuations than serious competitors, inflating the effective $N$ in the beta inversion.

The panel bound diagnostic motivates an investigation of effective competition. We estimate the “effective $N$” using four complementary methods in Appendix (ref): (i) counting bidders whose maximum bid exceeds 50% of the winning bid (median $N_{\text{eff}} = 8$), (ii) counting active final-stage bidders ($N_{\text{eff}} \approx 5$, limited by timestamp coarseness), (iii) estimating $N$ from order statistic spacing ($N_{\text{eff}} = 6$--$7$), and (iv) counting bidders within the top 90% of the bid range ($N_{\text{eff}} = 10$--$11$). All four methods broadly point to $N_{\text{eff}} \approx 5$--$10$, suggesting that roughly half of the unique bidders are casual participants. Re-estimating the lower bound with $N_{\text{eff}}$ from Methods 1 and 3 nearly eliminates the crossing with the panel upper bound, while the qualitative shape of the bounds is preserved. The bound width increases from 11.7--18.1 percentage points (baseline) to 15--29 percentage points ($N_{\text{eff}}$-adjusted), reflecting the reduced information from fewer effective competitors. The baseline results using actual $N_a$ should therefore be interpreted as an aggressive estimate, with the $N_{\text{eff}}$-adjusted bounds providing a conservative alternative.

Counterfactual Analyses

A key practical question is whether bounds on the valuation distribution can support meaningful policy analysis. This section answers in the affirmative through three exercises that translate the bounds $[F_L, F_U]$ into bounds on quantities of direct interest to sellers, platform designers, and regulators. The first quantifies how expected revenue depends on the continuation probability of future auctions in the Korean wholesale market: this is the empirical counterpart to the paper's central theoretical mechanism, the option to wait. The second asks how Cars and Bids seller revenue would change if the platform converted casual browsers into serious competitors, providing a price tag on bidder-engagement mechanisms such as qualifying deposits or activity-based filters. The third investigates the reserve-price envelope to identify a robust set of reserve choices that do not require committing to any particular $F$ within the bounds. Because each counterfactual reduces to an order-statistic functional of $F$, the bounds on $F$ map directly into bounds on the policy quantity, with no equilibrium or distributional assumptions beyond those already maintained.

The key building block is the expected order statistic: for $N$ i.i.d.\ draws from $F$, the $j$th highest order statistic has CDF $G_{j:N}(v;F) = I_{F(v)}(N\!-\!j\!+\!1, j)$. Since $I_z(a,b)$ is increasing in $z$, $F_L \leq F \leq F_U$ implies

where $E_F[\theta^{j:N}] = \underline{\theta} + \int_{\underline{\theta}}^{\bar{\theta}} [1 - G_{j:N}(v;F)]\, dv$. For the winner rent, we use the identity $E_F[\theta^{1:N} - \theta^{2:N}] = \int_{\underline{\theta}}^{\bar{\theta}} N F(v)^{N-1}(1 - F(v))\, dv$. Since the integrand depends only on $F(v)$ pointwise, bounds follow from minimizing and maximizing the integrand over $F(v) \in [F_L(v), F_U(v)]$ at each $v$ (which has a closed form since $N F^{N-1}(1-F)$ is unimodal in $F$ with maximum at $F^* = (N-1)/N$).

Future-Auction Uncertainty in the Korean Market

Sequential auction markets are routinely subject to disruptions that threaten the continuity of supply: weather and logistics shocks at wholesale yards, regulatory pauses, dealer strikes, and shifts in consignment volume can all reduce or eliminate the next sale opportunity. These disruptions are not benign supply-side events; by changing what bidders expect of future auctions, they alter the competitive dynamics of the current sale. In our framework, the option value of waiting is what depresses the current-period transaction price relative to a static benchmark, and when the option to wait is curtailed, they bid more aggressively today, raising current revenue at the cost of foregone future revenue. Quantifying this trade-off is essential for any platform that wishes to evaluate the revenue cost of a delay, the value of guaranteeing a future sale schedule, or the design of cancellation and rescheduling policies.

To formalize the exercise, we perturb the continuation probability $\tau \in [0,1]$: with this probability, the next auction in the observed sequence occurs, and with probability $1-\tau$, the future opportunity disappears. The case $\tau = 1$ corresponds to the certainty of a next auction (bidders fully internalize the option to wait), while $\tau = 0$ corresponds to a static auction with no continuation value. We implement the counterfactual on Korean auction-day pairs. For each consecutive pair of auctions $a$, with estimated common values $\hat\phi_{1a} > \hat\phi_{2a}$ and market competition $N_a$, the two-period benchmark model implies

The counterfactual therefore does not require simulating bid paths under a new dynamic equilibrium: the equilibrium price formula reduces the exercise to bounded order-statistic expectations, evaluated at the category-specific estimates of $F_L$ and $F_U$. We average the resulting bounds across observed descending pairs with $N_a \geq 3$.

A particularly clean object that emerges from this exercise is the waiting discount: the difference between first-period revenue under no future auction ($\tau = 0$) and first-period revenue when the future auction occurs with probability $\tau$. Subtracting $E[P_1(\tau)] = (\hat\phi_1 - \tau\hat\phi_2) E[\theta^{2:N}] + \tau\hat\phi_2 E[\theta^{3:N}]$ from $E[P_1^{\text{static}}] = \hat\phi_1 E[\theta^{2:N}]$ gives, after cancellation,

This decomposition isolates the option-value mechanism: the discount is exactly proportional to the gap between the second- and third-highest order statistics, scaled by $\tau\hat\phi_2$. Because both revenue components and the waiting discount are affine in $\tau$, Table (ref) reports the full counterfactual content on a compact grid of continuation probabilities; values between grid points are obtained by linear interpolation. Moving from $\tau = 0$ to $\tau = 1$ lowers the first-period lower envelope by 8.7--10.9% and the upper envelope by 8.1--10.0% across the five categories, while total expected revenue over the two-auction window rises by 12.9--29.2% on the lower envelope and 15.2--30.3% on the upper envelope. The waiting discount at $\tau=1$ ranges from 4.3--55.5 on the lower bound to 47.7--94.4 on the upper bound, in units of 10{,}000 KRW, with the largest discount in the Manual Non-Sedan Diesel segment where the order-statistic gap is widest.

The option-value effect places a lower bound on the cost of supply disruptions: in the Korean market, an unanticipated cancellation of the next sale would raise current-period revenue by about 9--12%, but the platform loses the entire next-period revenue stream, yielding a net loss of 11--23% on the two-auction window. Sellers and platforms can therefore use the bounds to value insurance, hedging contracts, or guaranteed-schedule mechanisms that reduce $1-\tau$. The asymmetry between the period-1 gain and the total loss shows that the price increase from supply tightening is small relative to the missing future revenue, so aggressive policies that artificially restrict supply to raise current prices are unambiguously revenue-reducing in this framework.

Effective Competition and Seller Revenue (Cars and Bids)

Online auction platforms typically report headline participation numbers based on registered bidders, page views, or watchlist additions. These metrics overstate the competitive intensity that actually determines transaction prices, because many participants are casual browsers who never bid seriously, and the platform's revenue is driven by the small subset of bidders who push prices toward valuations as we documented in the previous section. A natural follow-up question is the policy-relevant magnitude of this gap: what would seller revenue look like if the platform attracted, say, twice as many serious bidders? This question is operationally meaningful because platforms have direct levers for converting casual browsers into active bidders, with qualifying deposits, late-bidding reminders, anti-sniping extensions, and curated notifications being prominent examples. To answer the question, in the Cars and Bids application, we sweep a counterfactual number of serious bidders $N' \in \{5, 8, 10, 14, 20, 30, 50\}$ and compute bounds on seller revenue $\phi \cdot E[\theta^{2:N'}]$ and on winner rent $\phi \cdot E[\theta^{1:N'} - \theta^{2:N'}]$, holding the within-cluster valuation distribution within the estimated bounds. The first quantity uses the order-statistic bounds in (ref) with $j = 2$; the second uses the closed-form bound on the rent integrand $N'F(v)^{N'-1}(1-F(v))$ over $F(v) \in [F_L(v),F_U(v)]$. The maintained assumption is that the additional bidders draw from the same valuation distribution as the existing serious bidders.

Figure (ref) displays the resulting bounds. Moving from $N' = 8$ (the main effective-competition estimate) to $N' = 20$ raises the lower seller-revenue envelope by 40--46% across clusters and the upper envelope by 36--65%, while winner rents move in the opposite direction: in Cluster 1, for example, the rent interval falls from $[0.23,0.68]$ at $N'=8$ to $[0.10,0.53]$ at $N'=20$, a roughly 25--60% compression. These two patterns are economically symmetric: thicker competition redistributes surplus from winners to sellers without changing the value of the underlying allocation, which is the textbook prediction of standard IPV theory and which our bounds confirm in magnitude. The magnitudes provide a price tag for engagement mechanisms: the lower-bound revenue increase by raising effective competition from 8 to 20 bidders, 40--46% per cluster, can be compared directly with the cost of any platform investment that achieves that increase, such as marketing spend, deposit requirements, or anti-sniping rule changes. An investment whose cost is below this lower-bound revenue gain would be profitable in this partial-equilibrium calculation across all valuation distributions consistent with the data. The rent compression matters for incentive design, since as competition thickens, winner rents fall and the platform's ability to extract surplus through reserve prices, fees, or membership tiers shifts accordingly. A platform with a fee structure calibrated to a low-competition regime may leave revenue on the table as competition grows.

Reserve Price Envelope (Cars and Bids)

Reserve pricing is the most direct policy lever available to a seller in an English auction since the reserve sets a floor on the realized transaction price. Classical optimal-auction theory delivers a sharp prescription: raise the reserve until the marginal cost of foregone trade equals the marginal revenue from the reserve payment. However, this prescription requires committing to a particular valuation distribution $F$. In our partial identification framework, there is no unique optimal reserve identified. What replaces the point optimum is a set of reserves and an envelope of revenue functions, one for each $F$ within the bounds. Two natural objects of interest are the lower envelope, which describes the worst-case revenue at each reserve, and the maximin reserve, which maximizes the worst case. Both are interpretable without further assumptions about which $F$ in $[F_L,F_U]$ is correct, and both are directly actionable by a seller who wishes to hedge against misspecification.

Let $\rho = r/\phi$ denote the normalized reserve. Under a static second-price auction with $N$ i.i.d.\ bidders drawing from $F$, expected normalized revenue is

where the first term is the expected reserve payment in the event that exactly one bidder clears $\rho$ (so the price equals the reserve) and the integral is the expected second-highest valuation above $\rho$ (which determines the price when at least two bidders clear). The integral is monotone decreasing in $F$ and is bounded by evaluating it at $F_U$ and $F_L$. The reserve-payment term is non-monotone in $F$ pointwise (it is unimodal with maximum at $F^* = (N-1)/N$) so its bounds are obtained by minimizing and maximizing the integrand over $F(\rho) \in [F_L(\rho), F_U(\rho)]$ at each $\rho$. Combining the two terms gives pointwise revenue bounds on $\text{Rev}(\rho; F, N)$ for each $\rho$. We sweep $\rho \in [0.5, 2.5]$ at the effective-competition estimate $N = 8$.

Figure (ref) displays the revenue envelope for each cluster. The lower envelope is bimodal in every cluster: it attains a boundary value at $\rho = 0.5$ that is driven almost entirely by the integral term, drops as $\rho$ rises and the integral term collapses faster than the reserve-payment term builds up, and recovers to an interior local maximum near $\rho \approx 1.3$--$1.4$ where the reserve-payment term contributes more than 95% of expected revenue. Which of these two local maxima dominates varies across clusters. In Clusters 2--4 the interior maximum at $\rho^* = 1.3$, $1.3$, and $1.4$ exceeds the boundary value by 14.6%, 23.0%, and 15.6% respectively, so the maximin reserve is interior and meaningful. In Clusters 1 and 5 the two local maxima are essentially tied (the boundary wins by 0.6% and 4%, respectively) so the maximin reserve formally sits at $\rho^* = 0.5$, but a near-identical worst-case revenue is achievable at the interior local maximum near $\rho \approx 1.3$--$1.4$. The upper envelope, by contrast, is maximized at $\rho = 0.5$ in every cluster: under the most favorable distributions within the bounds, $F_U$ rises rapidly above $\rho \approx 1$ and the integral term dominates throughout. The lower and upper envelopes prescribe substantively different reserves in three of the five clusters, and even in the two clusters where they nominally agree at $\rho^* = 0.5$ the lower envelope identifies a near-tie interior local maximum.

This exercise provides useful guidance on reserve price policies. First, the divergence between the maximin and the upper-envelope optima in Clusters 2--4 reveals genuine distributional ambiguity: a planner who is confident that $F$ lies near the upper bound prefers a low reserve, while one who hedges against the worst case prefers a moderate reserve of $\rho^* \approx 1.3$--$1.4$. The choice between these two policy stances is a value judgment about ambiguity aversion, not an inference question that more data can settle. Second, the cluster-level heterogeneity is itself a finding: vehicle segments do not share a common optimal reserve, and a platform-wide uniform reserve policy can leave revenue gains relative to a cluster-specific policy in three of five clusters. Even under the most conservative robustness criterion, segment-targeted reserves can raise revenue by 15--23% relative to the boundary policy that a one-size-fits-all maximin would prescribe.

Conclusion

This paper develops a partial identification approach for the distribution of bidders' valuations in sequential English auctions. Our framework relies on two behavioral restrictions: bidders do not overbid and they account for future auction opportunities, without specifying any particular equilibrium. Our approach has several advantages over existing methods. First, it is robust to a variety of nonstandard features that cause equilibrium-based methods to fail, including supply uncertainty, loss aversion, and non-equilibrium bidding. Second, simulation experiments demonstrate that ignoring the sequential structure leads to misspecified bounds, while our method provides informative bounds that are valid under the maintained assumptions. Third, the method uses non-terminal as well as terminal auction information, yielding tighter bounds than last-period-only approaches when common-value components decline substantially. Two empirical applications demonstrate that the method is practical and yields informative bounds with real data.

From a practical standpoint, the bounds on $F$ characterize the distribution of buyer valuations without imposing equilibrium assumptions, providing a robust foundation for auction design. For the Cars and Bids platform, the effective-competition analysis suggests that a substantial share of observed bidders are casual participants, so mechanisms to distinguish active from passive bidders may improve price discovery. More broadly, the bounds on $F$ across market segments serve as inputs to counterfactual policy analysis, including the evaluation of supply uncertainty, effective competition, and reserve price rules. Computing optimal dynamic reserve policies in sequential auctions remains challenging because the interaction between current reserves and future bidder behavior requires equilibrium assumptions that our framework deliberately avoids. Developing such counterfactual analyses within a partial identification framework is an important direction for future work. Other avenues for future research include extending the approach to settings with unobserved heterogeneity and multi-unit demand.